Is it possible to combine spline IK for the body and regular IK for limbs?
as far as your question goes it is possible, it has to be set-up correctly - Where is your 'Spline IK' ??


  I would like the spine to move along the spline curve in the
  background, following its curvature


That would be a 'Path' and could be done several ways 'Clamp To' or 'Path' constraint. The set-up on the legs does not look good to me, your leg bones with the IK should be shifted forward to what is prob the current IK targets and a new or that large bone re=scaled to be the IK targets. That is why the legs are 'Glitching out' on you. I will try and mock up a file for you if you want.
DISCLAIMER: This file created in v2.76 - If using 2.7x it will open fine,, If using 2.8x 'APPEND' the objects from the 'Object' folder.